Galois group: 24T33

• Label: 24T33
• Degree: $24$
• Order: $48$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $D_6:C_4$

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$33$
Group:		$D_6:C_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		(3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,15,6,19,9,24,13,3,17,7,21,11)(2,16,5,20,10,23,14,4,18,8,22,12)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$12$:	$D_{6}$
$16$:	$C_2^2:C_4$
$24$:	$S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 6: $D_{6}$

Degree 8: $C_2^2:C_4$

Degree 12: $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Low degree siblings

24T33

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{12}$	$1$	$2$	$12$	$( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23)(12,24)$
2B	$2^{12}$	$1$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2C	$2^{12}$	$1$	$2$	$12$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24)(12,23)$
2D	$2^{10},1^{4}$	$6$	$2$	$10$	$( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$
2E	$2^{12}$	$6$	$2$	$12$	$( 1,21)( 2,22)( 3, 8)( 4, 7)( 5,18)( 6,17)( 9,13)(10,14)(11,23)(12,24)(15,20)(16,19)$
3A	$3^{8}$	$2$	$3$	$16$	$( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24)( 8,16,23)$
4A1	$4^{6}$	$2$	$4$	$18$	$( 1,19,13, 7)( 2,20,14, 8)( 3,21,15, 9)( 4,22,16,10)( 5,23,18,12)( 6,24,17,11)$
4A-1	$4^{6}$	$2$	$4$	$18$	$( 1, 7,13,19)( 2, 8,14,20)( 3, 9,15,21)( 4,10,16,22)( 5,12,18,23)( 6,11,17,24)$
4B1	$4^{6}$	$6$	$4$	$18$	$( 1,23,14,11)( 2,24,13,12)( 3, 9,16,22)( 4,10,15,21)( 5,19,17, 8)( 6,20,18, 7)$
4B-1	$4^{6}$	$6$	$4$	$18$	$( 1,20,14, 7)( 2,19,13, 8)( 3, 6,16,18)( 4, 5,15,17)( 9,12,22,24)(10,11,21,23)$
6A	$6^{4}$	$2$	$6$	$20$	$( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
6B	$6^{4}$	$2$	$6$	$20$	$( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
6C	$6^{4}$	$2$	$6$	$20$	$( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
12A1	$12^{2}$	$2$	$12$	$22$	$( 1,11,21, 7,17, 3,13,24, 9,19, 6,15)( 2,12,22, 8,18, 4,14,23,10,20, 5,16)$
12A-1	$12^{2}$	$2$	$12$	$22$	$( 1, 3, 6, 7, 9,11,13,15,17,19,21,24)( 2, 4, 5, 8,10,12,14,16,18,20,22,23)$
12A5	$12^{2}$	$2$	$12$	$22$	$( 1,24,21,19,17,15,13,11, 9, 7, 6, 3)( 2,23,22,20,18,16,14,12,10, 8, 5, 4)$
12A-5	$12^{2}$	$2$	$12$	$22$	$( 1,15, 6,19, 9,24,13, 3,17, 7,21,11)( 2,16, 5,20,10,23,14, 4,18, 8,22,12)$

Malle's constant $a(G)$:     $1/10$

Group invariants

Order:		$48=2^{4} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		48.14
Character table:

		1A	2A	2B	2C	2D	2E	3A	4A1	4A-1	4B1	4B-1	6A	6B	6C	12A1	12A-1	12A5	12A-5
	Size	1	1	1	1	6	6	2	2	2	6	6	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	3A	2C	2C	2A	2A	3A	3A	3A	6C	6C	6C	6C
	3 P	1A	2A	2B	2C	2D	2E	1A	4A-1	4A1	4B-1	4B1	2A	2B	2C	4A-1	4A-1	4A1	4A1
	Type
48.14.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.14.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
48.14.1c	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.14.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
48.14.1e1	C	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-i$	$i$	$i$	$-i$	$-1$	$1$	$-1$	$i$	$-i$	$i$	$-i$
48.14.1e2	C	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$i$	$-i$	$-i$	$i$	$-1$	$1$	$-1$	$-i$	$i$	$-i$	$i$
48.14.1f1	C	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-i$	$i$	$-i$	$i$	$-1$	$1$	$-1$	$i$	$-i$	$i$	$-i$
48.14.1f2	C	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$i$	$-i$	$i$	$-i$	$-1$	$1$	$-1$	$-i$	$i$	$-i$	$i$
48.14.2a	R	$2$	$2$	$2$	$2$	$0$	$0$	$-1$	$2$	$2$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
48.14.2b	R	$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$
48.14.2c	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$
48.14.2d	R	$2$	$2$	$2$	$2$	$0$	$0$	$-1$	$-2$	$-2$	$0$	$0$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
48.14.2e1	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$-1$	$-2i$	$2i$	$0$	$0$	$1$	$-1$	$1$	$-i$	$i$	$-i$	$i$
48.14.2e2	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$-1$	$2i$	$-2i$	$0$	$0$	$1$	$-1$	$1$	$i$	$-i$	$i$	$-i$
48.14.2f1	R	$2$	$-2$	$2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$0$	$1$	$1$	$-1$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$
48.14.2f2	R	$2$	$-2$	$2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$0$	$1$	$1$	$-1$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$
48.14.2g1	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$
48.14.2g2	C	$2$	$2$	$-2$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$